Improved quantum lower and upper bounds for matrix scaling

by   Sander Gribling, et al.

Matrix scaling is a simple to state, yet widely applicable linear-algebraic problem: the goal is to scale the rows and columns of a given non-negative matrix such that the rescaled matrix has prescribed row and column sums. Motivated by recent results on first-order quantum algorithms for matrix scaling, we investigate the possibilities for quantum speedups for classical second-order algorithms, which comprise the state-of-the-art in the classical setting. We first show that there can be essentially no quantum speedup in terms of the input size in the high-precision regime: any quantum algorithm that solves the matrix scaling problem for n × n matrices with at most m non-zero entries and with ℓ_2-error ε=Θ(1/m) must make Ω(m) queries to the matrix, even when the success probability is exponentially small in n. Additionally, we show that for ε∈[1/n,1/2], any quantum algorithm capable of producing ε/100-ℓ_1-approximations of the row-sum vector of a (dense) normalized matrix uses Ω(n/ε) queries, and that there exists a constant ε_0>0 for which this problem takes Ω(n^1.5) queries. To complement these results we give improved quantum algorithms in the low-precision regime: with quantum graph sparsification and amplitude estimation, a box-constrained Newton method can be sped up in the large-ε regime, and outperforms previous quantum algorithms. For entrywise-positive matrices, we find an ε-ℓ_1-scaling in time O(n^1.5/ε^2), whereas the best previously known bounds were O(n^2polylog(1/ε)) (classical) and O(n^1.5/ε^3) (quantum).


page 1

page 2

page 3

page 4


Quantum algorithms for matrix scaling and matrix balancing

Matrix scaling and matrix balancing are two basic linear-algebraic probl...

Spectral sparsification of matrix inputs as a preprocessing step for quantum algorithms

We study the potential utility of classical techniques of spectral spars...

Scaling positive random matrices: concentration and asymptotic convergence

It is well known that any positive matrix can be scaled to have prescrib...

n-Qubit Operations on Sphere and Queueing Scaling Limits for Programmable Quantum Computer

We study n-qubit operation rules on (n+1)-sphere with the target to help...

Improved estimates for the number of non-negative integer matrices with given row and column sums

The number of non-negative integer matrices with given row and column su...

Better and Simpler Error Analysis of the Sinkhorn-Knopp Algorithm for Matrix Scaling

Given a non-negative n × m real matrix A, the matrix scaling problem is...

Quantum query complexity with matrix-vector products

We study quantum algorithms that learn properties of a matrix using quer...

Please sign up or login with your details

Forgot password? Click here to reset